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KVPY2020MathematicsApplication of Derivatives

Let R be the set of all real numbers and f ( x ) = sin 10 x cos 8 x + cos 4 x + cos 2 x + 1 for x ∈ R . Let S = λ ∈ R ∣ there exits a point c ∈ ( 0 , 2 π ) with f ' ( c ) = λ f ( c ) Then

Options

  1. AS = R
  2. BS = 0
  3. CS = [ 0 , 2 π ]
  4. DS is a finite set having more than one element

Correct answer

A. S = R

Step-by-step solution

Let g ( x ) = f ( x ) e - λ x ; x ∈ [ 0 , 2 π ] so, g ( 0 ) = g ( 2 π ) = 0   as   f ( 0 ) = f ( 2 π ) = 0 Thus, ∃ c ∈ ( 0 , 2 π ) such that g ' ( c ) = 0 ⇒ f ' ( c ) = λ f ( c ) ∀ λ ∈ R ⇒ S = R

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